Some details

Notice that for Σ\Sigma an (n+1)(n+1)-dimensional manifold with nn-dimensional boundary∂Σ\partial \Sigma, regarded as a cobordismΣ:∅→∂Σ\Sigma : \emptyset \to \partial \Sigma, an (n+1)(n+1)-dimensional TQFT assigns a morphism

Z(Σ):1→Z(∂Σ),
Z(\Sigma) : 1 \to Z(\partial \Sigma)
\,,

hence an element of the space Z(∂Σ)Z(\partial \Sigma). Under holography, this element is identified with the partition function of an nn-dimensional QFT evaluated on the manifold (without boundary) ∂Σ\partial \Sigma.

Remark

In view of these two classes of examples it is maybe noteworthy that one can see that also closed string field theory, which is supposed to be one side of the AdS/CFT correspondence, has the form of an infinity-Chern-Simons theory, as discussed there, for the L-infinity algebra of closed string correlators. So maybe the above two different realizations of the holographic principle are really aspects of one single mechanism for ∞\infty-Chern-Simons theory.

a correlator for the WZW model with source field AA has to satisfy a conformal transformation property called a Ward identity. The space of all suitable functionals satisfying these identities is the space of conformal blocks. That space is equivalently identified with the space of wave functions of Chern-Simons theory depending on the fields AA, hence the quantum states of the CS theory.

More generally, consider some nn-dimensional FQFTZBZ_B and assume that the spaces of states that it assigns to any (n−1)(n-1)-dimensional manifold XX are of finite dimension (over some ground field ℂ\mathbb{C}):

dimZB(X)<∞.
dim Z_B(X) \lt \infty
\,.

Then for Σ:∂inΣ→∂outΣ\Sigma : \partial_{in}{\Sigma} \to \partial_{out}{\Sigma} any cobordism of dimension nn, the correlator

Stated differently: the vector space ZB(∂Σ)Z_B(\partial \Sigma) is the space of all “potential correlators” of ZBZ_B and ZB(Σ)¯\overline{Z_B(\Sigma)} is the particular one chosen by the given model.

If ZBZ_B is really a CFT one calls a subspace BlB(Σ)⊂Z(∂Σ)Bl_B(\Sigma) \subset Z(\partial\Sigma) of elements that respect conformal invariance in a certain way the space of conformal blocks and calls the assignment Σ↦BlB(Σ)\Sigma \mapsto Bl_B(\Sigma) the modular functor of the model.

Notice that by looking at all “potential correlators” this way we are suddenly assigning vector spaces in codimension 0 (on Σ\Sigma), even though the axioms of an FQFT a priori only mention vector spaces (of states) assigned in codimension 1. Given all these spaces of “conformal blocks”, the (re)construction of ZBZ_B consists of choosing inside each BlB(Σ)Bl_B(\Sigma) the actual correlator ZB(Σ)¯\overline{Z_B(\Sigma)} (this way of looking at TQFTs BB is actually the way in which Atiyah originally formuated the axioms of FQFT).

But since we are dealing now with vector spaces assigned to nn-dimensional Σ\Sigma, we can ask the following question:

Notice that ZA(Σ)Z_A(\Sigma) is the space of states of AA over Σ\Sigma, while BlB(Σ)Bl_B(\Sigma) is the space of possible correlators of BB over Σ\Sigma. Under holography, the states of AA are identified with the correlators of BB.

Holography of higher Chern-Simons/CFT-type

RT-3d TQFT / rational 2d CFT

The class of examples of “Chern-Simons-type holography” we mention now has fairly completely and rigorously been understood. It is in turn a special and comparatively simple (but far from trivial) case of the historically earliest class of examples: ordinary Chern-Simons theory dual to a 2d WZW model below.

At the level of action functionals the relation is directly seen by observing that on a 3-d manifold with boundary the Chern-Simons theory action is not gauge invariant, but has a boundary term depending on the gauge transformation. Since the gauge transformation is a function on the 2d boundary with values in GG, this boundary term is like an action functional for a sigma-model with target spaceGG, and indeed it is that (subject to some fine-tuning) of the GG-WZW model.

Therefore indeed the symplectic pairing vanishes on the self-dual and on the anti-selfdual forms. Evidently these provide a decomposition into Lagrangian subspaces.

Therefore a state of higher Chern-Simons theory on Σ\Sigma may locally be thought of as a function of the self-dual forms on Σ\Sigma. Under holography this is (therefore) identified with the correlator of a self-dual higher gauge theory on Σ\Sigma.

Chern-Simons/CFT in AdS/CFT

it is argued that in the AdS/CFT correspondence it is in fact just the Chern-Simon terms inside the corresponding supergravity theories whose states control the conformal blocks of the dual CFT. So the CS/CFT correspondence is a part (a crucial part) of the AdS/CFT correspondence, at least for AdS5/CFT4AdS_5/CFT_4 and AdS7/CFT6AdS_7/CFT_6.